"Glad to be back with my dawg," Mahomes said on Twitter. "Welcome to #ChiefsKingdom"
Yahoo Sports was the first to report Washington's agreement with the Chiefs.
Chiefs starting running back Damien Williams returns, but the team doesn't have much depth behind him. Other backs under contract are Darrel Williams, whose 2019 season ended in December because of a torn hamstring, and Darwin Thompson. Both played sparingly.
The Raiders chose to re-sign running back Jalen Richard over Washington because of Richard's pass-catching ability.
Washington acquitted himself well in rookie sensation Josh Jacobs' absence down the stretch with the Raiders last season.
With Jacobs missing three of the Raiders' final four games with a fractured shoulder, Washington assumed the role of bell-cow back and had a combined 334 yards of offense and two touchdowns in those three games (rushing for 215 yards and those two scores on 54 carries and catching 16 passes for 119 yards).
He was especially impressive in a Week 16 victory at the Los Angeles Chargers, carrying the ball 23 times for 85 yards (the second-highest single-game total of his career).
Washington, 27, is coming off a season in which he played in 16 games for the first time in his career, with career highs in starts (3), carries (108) and receptions (36).
At just 5-foot-8, 210 pounds, Washington is a tenacious runner. He has rushed for 1,122 yards on 282 carries (4.0 ypc) in his career with seven touchdowns and has caught 88 passes for 613 yards and a score.
ESPN's Adam Teicher and Paul Gutierrez contributed to this report.